OPTIMAL TRANSPORT OF LOW ENERGY PARTICLE BEAMS*Christopher K. Allen, Samar K. Guharay, and Martin Reiser
Institute for Plasma Research
University of Maryland, College Park, MD 20742
The transport and matching problem for a low energy
transport system is approached from a control theoretical
viewpoint. We model the beam dynamics and transport
section using the KV envelope equations. To this model weapply the principles of optimal control to formulatetechniques which aid in the design of the transport andmatching section. The techniques are applied to theexample of an H- beam transport and matching system.

I. INTRODUCTIONThe design of particle beam transport and matchingsystems has typically been accomplished in much the same
way an experiment is run. A computer program is used tosimulate the behavior of the beam in a given transportsystem. The knobs of this simulated system are thenadjusted until a satisfactory solution is obtained. This canbe a lengthy and arduous process. The progress of such a
procedure relies completely upon the experience,
judgement, and intuition of the designer.
It is the goal of this work to utilize the principles of
optimal control theory to aid in the design of beamtransport and matching systems. We have developed anautomated technique which determines the optimal lens
strengths to match the beam envelope to a prescribed finalstate. In this paper, we consider beams with ellipticalsymmetry and apply the results to example Low Energy
Beam Transport (LEBT) sections where space charge plays
a dominant role.

II. MATHEMATICAL MODEL

X" +κ x ( z ) X −(1)where the prime indicates differentiation with respect to z,
K is the generalized beam perveance, and ε x and ε y are theeffective emittances of the beam in the x and y planes,
respectively. The functions κ x(z) and κ y(z) represent the* Supported by DOE and ONR

B. The Transport System
The physical transport section consists of N discrete
focusing lenses cascading axially. One of the mostimportant assumptions we make in the paper is that theaction of each lens is independent of the others. That is to
say that while the beam propagates through a lens, no otherlens affects it. This is not always physical, since we know
that, in the case of electrostatic or magnetostatic
quadrupole lenses, the fields of one lens tend to leak into
the regions of any adjacent lenses. Typically, however, wemay neglect any small coupling of this type and still aquire
accurate results. Consequently, for each lens the focusing
function is nonzero only on a finite interval of the z axis.
C. Boundary Conditions (Matching)
We are given initial conditions for the beam envelope
at the transport section's entrance position, z=zi . Labelthese initial conditions (X j,X i') for the x plane and (Y j,Y i ')for the y plane. In the case of a transport and matchingsystem we are also given desired final conditions at z=zf ,the exit location of the transport section. We will callthese conditions (X f,X f') and (Y f,Y f') for the x and y planes,
respectively. Thus, we must satisfy the following boundary
conditions along with Eq. (1):( )X' ( )X' ( z f ) = X 'f ,Y (z f ) = Yf ,Y ( zi ) = Yi ,Y' ( zi ) = Yi' , Y' ( z f ) = Y f' ,X ( zi ) = Xi ,A. Beam Dynamics
We model the particle beam using the KV
envelope equations. In the two-dimensional steady-statecase these equations model a uniform density beam withelliptical cross-section. Let X(z) and Y(z) represent thebeam envelope semi-axes in the x and y planes,respectively. This system may be described by the systemof coupled differential equations [1]ε 22K− x3 = 0,
X +Y Xε y22K− 3 = 0,Y " +κ y ( z )Y −X +Y Yaction of the transport section in the x and y planes,respectively. They are usually referred to as the focusing orcontrol functions. These equations also describe thebehavior of the r.m.s. beam envelope for any beam withelliptical symmetry in the xy plane [1,2].

X zf = Xf ,zi = Xi' ,D. Cost Functional
Let X(z) and Y(z) represent the x and y envelopes ofsome desirable reference trajectory for the system (thesefunctions would be chosen by the designer). If the pair[X(z),Y(z)] is the actual solution to Eq. (1) for a given
system, then a plausible merit functional J for the solution
trajectory is given byzf() ()221
J[ X ( z ), Y ( z )] ≡ ∫  X ( z ) − X ( z ) + Y ( z ) − Y ( z )  dz.
2 z i(3)This functional, in essence, measures the distance
between the solution trajectory [X(z),Y(z)] and the reference
trajectory [X(z),Y(z)].
The boundary conditions of Eq.'s (2) may be included in
the functional J with the addition of a boundary term Φ[ ( ) ( )] ≡ 12 [ X(z ) − X ] + 12 [Y (z ) − Y ] .

Φ X z f ,Y z f

2ff

2ff

set of starting values for the un’s. Once started, thealgorithm will pick out local minima in the vicinity of thisstarting set. This is quite unlike dynamic programming,
which is a global technique not requiring anydifferentiability conditions.

IV. EXAMPLE(4)

Thus, the functional which we actually minimize isJ[ X ( z ), Y ( z )] + Φ X z f , Y z f .

[ ( ) ( )]III. OPTIMIZATION TECHNIQUE
Since the lens actions are independent, the focusing
function κ (z) may be subsectioned into N discrete parts,
one for each lens. Denote the focusing function for lens n asκ n(z). Also the functional profile for each κn(z) is known
from the geometry of the lens. Therefore, only theamplitude of each κn(z) remains variable (we do not vary
the axial placement of each lens). Denotetheseamplitudes u n . We are left with a linear cascade of discretelenses which act on the beam, in succession, according toEq. (1). The beam is steered solely by adjusting the set ofcontrols {un}. This situation is referred to in the literature
as a multistage control network [3]. The formal control
problem is stated as: find the sequence of controls {un}which steers the system state from [Xi,Yi] to [Xf,Yf] according
to the dynamics of Eq. (1) and which minimizes the merit
functional of Eq. (3).
We employ two different techniques from optimal
control theory to solve this problem. The first is dynamic
programming which has been outlined in a previous paper[4]. The technique works well for axisymmetric systems
but usually becomes too CPU intensive for the twodimensional KV equations. Rather, in this situation notethat Eq. (1) yields X(z) and Y(z) as implicit functions of theun’s. Therefore the functional J may also be regarded as afunction of the un’s. The next logical step to this method ofrepresentation would be to take the gradient of J withrespect to the un’s. Once we have this gradient, we may use
nonlinear programming to search for the minimizing set oflens amplitudes [5]. This approach constitutes the second
technique for solving the control problem. Fortuitously, thiscontrol problem has a rich mathematical structure which
may be exploited for computation of the gradients. It ispossible to find them using only numerical integration,rather than differentiation. This yields a more accurate anda more stable search algorithm.
The major advantage of the second approach is that it issubstantially faster than dynamic programming. So much sothat the algorithm usually converges in a matter of minutes(dynamic programming for the fully two dimensional casetypically has run times on the order of a day). The major
disadvantage is that the technique searches out only local
minima. Consequently, it is necessary to pick a starting
point for the algorithm. That is, the designer must choose aBoth the algorithms discussed above have been
implemented in a computer-aided design program called
Spot, which runs on the PC under Microsoft Windows. It isan environment where the designer interacts with theoptimizer in order to steer it in the desired direction. In thisway the designer may quickly obtain local solutions to the
optimal control problem using the nonlinear programming
technique. Once found, the result may be checked using
dynamic programming.
A. LEBT System
We consider the case of a Low Energy Beam transport(LEBT) section for high-current, high-brightness H- beam
currently under study at the University of Maryland [6]. The
system is composed of six electrostatic quadrupole lenses(ESQ's) sandwiched between grounding shunts. We model
the action of each lens using the "hard-edge"
approximation. A detailed description of the system can befound in reference [6].
We list below the relevent
parameters.
ε x, ε yIV
X i & Y i X i ' & Yi '(mA)(keV) (m-rad)(mm)(mrad)
30

35

5.56x10 -5

1.25

50

Table 1: Beam Parameters
Lens No.'s
1&6
2&5
3&4

Aperature
Radius(mm)15.022.0
22.0

Length(mm)
25.0
59.0
47.0

Spacing(mm)
6.0
6.0
6.0

Table 2: LEBT Parameters
B. Design Guidelines
Our goal is to match the beam to the final state
Xf=Yf=1.25mm, Xf'=Yf'=-50mrad. We have the designguidelines that the beam excursions through the ESQs
should not exceeded 75% of the aperature radius; thisrequirement will minimize spherical aberrations. Also, thelens voltage seen at the beam envelope should not exceed
10% of the beam voltage in order to minimize chromaticaberrations. In the following figures the controls {un} areplotted, rather than the actual ESQ voltages. In ournonrelativistic situation, the conversion formula is given as
Vn = unVb a2(5)

where V n is the ESQ voltage, V b is the beam voltage and ais the ESQ aperature radius.

The reference trajectory we have chosen is a piecewise
linear function of z (we let X(z)=Y(z)=R(z) ).
0.00125 + 0.100 z if zε [0, 0.062]
R( z ) ≡ 0.01335
if zε [0, 0.062, 0.233](6)0.01335 − 0.100 z if zε [0.233, 0.354]
In this way the reference trajectory levels off to 60% of
the aperature radius of lenses 2,3,4, and 5 while remaining
well within the requirements of lenses 1 and 6. The
reference trajectory is shown in the following figures alongwith the corresponding solutions.
Figure 1 depicts the solution obtained by the nonlinear
programming technique without any constraints imposed
upon the lens voltages. The beam is essentially "bounced"
off of these two lenses. Clearly this is an unacceptable
solution since the beam envelope is comparable to the lensaperature. The situation is remedied by imposing
constraints on the lens voltages in the nonlinear
programming problem.
It was found that the current ESQ system cannot strictly
meet the criteria for minimization of chromatic aberrations.
A feasible solution was found when holding the lens
voltages seen at the beam envelope to 15% of the beam
voltage. This solution is shown in Figure 2. Note that thebeam focusing is distributed more evenly across the lenses.
We wish to compare these solutions with that obtained
previously without any automation. Figure 3 shows asolution obtained strictly by trial and error with the aim ofachieving the same design guidelines. Both show similar
characteristics. However, the trial and error solution
violates more of the design guidelines. The most notableviolations occur at the first lens, where the beam envelope
fills 85% of the ESQ aperature and the lens voltage at theenvelope is 20% of the beam voltage. This solution also
fails to meet the boundary conditions exactly; theconvergence is only -40mrad. We also mention that thesolution of Figure 3 was found over several hours by anexperienced designer while that of Figure 2 was found inless than half an hour.

Figure 1: Unconstrained Optimal Solution

Figure 2: Constrained Optimal Solution

V. CONCLUSION
The optimization techniques discussed here provide thebasis for a useful computer aided design tool. When
implemented as above, the designer may interactively
guide the optimizer to desirable solutions. These solutions
may then be checked against the dynamic programming
scheme. This saves a substantial amount of time in thedesign phase and also allows for the exploration of many
design alternatives.

VI. REFERENCES[1] M. Reiser, Theory and Design of Charged Particle
Beams (Wiley, 1994), Sect. 4.4.2, 5.3.4.
[2] F. R. Sacherer, "RMS Envelope Equations withSpace Charge", IEEE Trans. Nucl. Sci. NS-18, 1105 (1971).
[3] A.E. Bryson and Y.C. Ho, Applied Optimal Control(Wiley and Sons, 1975), Chapt. 2.

Figure 3: Trial and Error Solution[4] C.K. Allen, S.K. Guharay, and M. Reiser, "OptimalControl of Low Energy Particle Beams", Proceedings
Beams 94 Conf., San Diego, pp. 540-543.
[5] D. Luenberger, Linear and Nonlinear Programming:
Second Edition, (Addison-Wesley, 1984) Chapt.'s 8 and 9.
[6] S. K. Guharay, C. K. Allen, and M. Reiser, , Nucl.
Instr. and Meth. in Phys. Res., A 339 (1994), pp. 429-438.